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Mar 30, 2013 - Theorem 2.4. [4, Theorem 2.8] Let M be an R–module and n ≥ 0 an integer. Then there are exact sequences of functors: (2.4.1). 0 → Ext1.
arXiv:1210.5882v2 [math.AC] 30 Mar 2013

VANISHING OF COHOMOLOGY OVER COMPLETE INTERSECTION RINGS ARASH SADEGHI Abstract. Let R be a complete intersection ring and let M and N be R– modules. It is shown that the vanishing of ExtiR (M, N ) for a certain number of consecutive values of i starting at n forces the complete intersection dimension of M to be at most n−1. We also estimate the complete intersection dimension of M ∗ , the dual of M , in terms of vanishing of the cohomology modules, ExtiR (M, N ).

Contents 1. Introduction 2. Preliminaries 3. Vanishing of Ext for rigid modules 4. Vanishing of Ext over complete intersection rings References

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1. Introduction In this paper, we study the relationship between the vanishing of ExtiR (M, N ) for various consecutive values of i, and the complete intersection dimensions of M and M ∗ , the dual of M . The vanishing of homology was first studied by Auslander [3]. For two finitely generated modules M and N over an unramified regular local ring R R, he proved that if TorR i (M, N ) = 0 for some i > 0, then Torn (M, N ) = 0 for all i ≥ n. In [17], Lichtenbaum settled the ramified case. It is easy to see that a similar statement is not true in general, with Tor replaced by Ext. In [15], Jothilingam studied the vanishing of cohomology by using the rigidity Theorem of Auslander. For two nonzero modules M and N over a regular local ring R, he proved that if M satisfies (Sn ) for some n ≥ 0 and ExtiR (M, N ) = 0 for some positive integer i such that i ≥ depthR (N ) − n, then ExtjR (M, N ) = 0 for all j ≥ i. In [16], Jothilingam and Duraivel studied the relationship between the vanishing of ExtiR (M, N ) and the freeness of M ∗ . For two nonzero modules M and N over a regular local ring R, they proved that if ExtiR (M, N ) = 0 for all 1 ≤ i ≤ max{1, depthR (N ) − 2}, then M ∗ is free. In this paper we are going to generalize these results. Date: March 18, 2013. 2000 Mathematics Subject Classification. 13D07, 13H10. Key words and phrases. Complete intersection dimension, Complexity, Vanishing of cohomology, Grothendieck group. 1

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An R–module M is said to be c-rigid if for all R–modules N , TorR i+1 (M, N ) = R R TorR (M, N ) = · · · = Tor (M, N ) = 0 for some i ≥ 0 implies that Tor i+2 i+c n (M, N ) = 0 for all n > i. If c = 1 then we simply say that M is rigid. The aim of this paper is to study the following question. Question 1.1. Let R be a Gorenstein local ring and let M and N be R–modules such that N has reducible complexity. Assume that n ≥ 0, c > 0 are integers and that N is c-rigid. If ExtiR (M, N ) = 0 for all i, 1 ≤ i ≤ max{c, depthR (N ) − n}, then what can we say about the Gorenstein dimensions of M and M ∗ ? In section 2, we collect necessary notations, definitions and some known results which will be used in this paper. In section 3, we study the Question 1.1 for rigid modules. Over a Gorenstein local ring R, given nonzero R-modules M and N such that N has reducible complexity, we show that if N is rigid and ExtiR (M, N ) = 0 for all i, 1 ≤ i ≤ max{1, depthR (N ) − n} and some n ≥ 2, then G-dimR (M ∗ ) ≤ n − 2, which is a generalization of [16, Theorem 1]. In particular, if M satisfies (Sn ), then G-dimR (M ) = 0 (see Theorem 3.2). As a consequence, for two nonzero modules M and N over a complete intersection ring R, it is shown that if N is rigid and ExtiR (M, N ) = 0 for some positive integer i ≥ depthR (N ), then CI-dimR (M ) = sup{i | ExtiR (M, N ) 6= 0} < i (see Theorem 3.5). In section 4, we generalize [15, Corollary 1] for modules over a complete intersection ring. For two modules M and N over a complete intersection ring R with codimension c, it is shown that if M satisfies (St ) for some t ≥ 0, ExtiR (M, N ) = 0 for all i, n ≤ i ≤ n + c and some n > 0 and depthR (N ) ≤ n + c + t, then CI-dimR (M ) = sup{i | ExtiR (M, N ) 6= 0} < n (see Corollary 4.3). 2. Preliminaries Throughout the paper, (R, m) is a commutative Noetherian local ring and all modules are finite (i.e. finitely generated) R–modules. The codimension of R is defined to be the non-negative integer embdim(R) − dim(R) where embdim(R), the embedding dimension of R, is the minimal number of generators of m. Recall that b of R has the R is said to be a complete intersection if the m-adic completion R form Q/(f ), where f is a regular sequence of Q and Q is a regular local ring. A complete intersection of codimension one is called a hypersurface. A local ring R is b of R has said to be an admissible complete intersection if the m-adic completion R the form Q/(f ), where f is a regular sequence of Q and Q is a power series ring over a field or a discrete valuation ring. Let · · · → Fn+1 → Fn → Fn−1 → · · · → F0 → M → 0 be the minimal free resolution of M . Recall that the nth syzygy of an R–module M is the cokernel of the Fn+1 → Fn and denoted by Ωn M , and it is unique up to isomorphism. The nth Betti number, denoted βnR (M ), is the rank of the free R–module Fn . The complexity of M is defined as follows. cxR (M ) = inf{i ∈ N ∪ 0 | ∃γ ∈ R such that βnR (M ) ≤ γni−1 for n ≫ 0}. Note that cxR (M ) = cxR (Ωi M ) for every i ≥ 0. It follows from the definition that cxR (M ) = 0 if and only if pdR (M ) < ∞. If R is a complete intersection, then the complexity of M is less than or equal to the codimension of R (see [12]). The complete intersection dimension was introduced by Avramov, Gasharov and Peeva

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[6]. A module of finite complete intersection dimension behaves homologically like a module over a complete intersection. Recall that a quasi-deformation of R is a diagram R → A և Q of local homomorphisms, in which R → A is faithfully flat, and A և Q is surjective with kernel generated by a regular sequence. The module M has finite complete intersection dimension if there exists such a quasideformation for which pdQ (M ⊗R A) is finite. The complete intersection dimension of M , denoted CI-dimR (M ), is defined as follows. CI-dimR (M ) = inf{pdQ (M ⊗R A)−pdQ (A) | R → A և Q is a quasi-deformation }. The complete intersection dimension of M is bounded above by the projective dimension, pdR (M ), of M and if pdR (M ) < ∞, then the equality holds (see [6, Theorem 1.4]). Every module of finite complete intersection dimension has finite complexity (see [6, Theorem 5.3]). The concept of modules with reducible complexity was introduced by Bergh [7]. Let M and N be R–modules and consider a homogeneous element η in the graded L∞ |η| i R–module Ext∗R (M, N ) = i=0 ExtR (M, N ). Choose a map fη : ΩR (M ) → N representing η, and denote by Kη the pushout of this map and the inclusion |η| ΩR (M ) ֒→ F|η|−1 . Therefore we obtain a commutative diagram 0 −−−−→ Ω|η| M −−−−→ F|η|−1 −−−−→ Ω|η|−1 M −−−−→ 0     k f y y yη 0 −−−−→

N

−−−−→



−−−−→ Ω|η|−1 M −−−−→ 0.

with exact rows. Note that the module Kη is independent, up to isomorphism, of the map fη chosen to represent η. Definition 2.1. The full subcategory of R-modules consisting of the modules having reducible complexity is defined inductively as follows: (i) Every R-module of finite projective dimension has reducible complexity. (ii) An R-module M of finite positive complexity has reducible complexity if there exists a homogeneous element η ∈ Ext∗R (M, M ), of positive degree, such that cxR (Kη ) < cxR (M ), depthR (M ) = depthR (Kη ) and Kη has reducible complexity. By [7, Proposition 2.2(i)], every module of finite complete intersection dimension has reducible complexity. In particular, every module over a local complete intersection ring has reducible complexity. On the other hand, there are modules having reducible complexity but whose complete intersection dimension is infinite (see for example, [9, Corollarry 4.7]). The notion of the Gorenstein(or G-) dimension was introduced by Auslander [2], and developed by Auslander and Bridger in [4]. Definition 2.2. An R–module M is said to be of G-dimension zero whenever (i) the biduality map M → M ∗∗ is an isomorphism. (ii) ExtiR (M, R) = 0 for all i > 0. (iii) ExtiR (M ∗ , R) = 0 for all i > 0.

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The Gorenstein dimension of M , denoted G-dimR (M ), is defined to be the infimum of all nonnegative integers n, such that there exists an exact sequence 0 → Gn → · · · → G0 → M → 0 in which all the Gi have G-dimension zero. By [4, Theorem 4.13], if M has finite Gorenstein dimension, then G-dimR (M ) = depth R − depthR (M ). By [6, Theorem 1.4], G-dimR (M ) is bounded above by the complete intersection dimension, CI-dimR (M ), of M and if CI-dimR (M ) < ∞, then the equality holds. Let R be a local ring and let M and N be finite nonzero R-modules. We say the pair (M, N ) satisfies the depth formula provided: depthR (M ⊗R N ) + depth R = depthR (M ) + depthR (N ). The depth formula was first studied by Auslander [3] for finite modules of finite projective dimension. In [13], Huneke and Wiegand proved that the depth formula holds for M and N over complete intersection rings R provided TorR i (M, N ) = 0 for all i > 0. In [9], Bergh and Jorgensen generalize this result for modules with reducible complexity over a local Gorenstein ring. More precisely, they proved the following result: Theorem 2.3. [9, Corollary 3.4] Let R be a Gorenstein local ring and let M and N be nonzero R–modules. If M has reducible complexity and TorR i (M, N ) = 0 for all i > 0, then depthR (M ⊗R N ) + depth R = depthR (M ) + depthR (N ). We denote by G(R) the Grothendieck group of finite modules over R, that is, the quotient of the free abelian group of all isomorphism classes of finite R–modules by the subgroup generated by the relations coming from short exact sequences of finite R-modules. We also denote by G(R) = G(R)/[R], the reduced Grothendieck group. For an abelian group G, we set GQ = G ⊗Z Q. f

Let P1 → P0 → M → 0 be a finite projective presentation of M . The transpose of M , Tr M , is defined to be coker f ∗ , where (−)∗ := HomR (−, R), which satisfies in the exact sequence (2.1)

0 → M ∗ → P0∗ → P1∗ → Tr M → 0

and is unique up to projective equivalence. Thus the minimal projective presentations of M represent isomorphic transposes of M . Two modules M and N are called stably isomorphic and write M ≈ N if M ⊕ P ∼ = N ⊕ Q for some projective modules P and Q. Note that M ∗ ≈ Ω2 Tr M by the exact sequence (2.1). The composed functors Tk := Tr Ωk−1 for k > 0 introduced by Auslander and Bridger in [4]. If ExtiR (M, R) = 0 for some i > 0, then it is easy to see that Ti M ≈ ΩTi+1 M . We frequently use the following Theorem of Auslander and Bridger. Theorem 2.4. [4, Theorem 2.8] Let M be an R–module and n ≥ 0 an integer. Then there are exact sequences of functors: (2.4.1) n 2 0 → Ext1R (Tn+1 M, −) → TorR n (M, −) → HomR (ExtR (M, R), −) → ExtR (Tn+1 M, −), (2.4.2) n n R TorR 2 (Tn+1 M, −) → (ExtR (M, R) ⊗R −) → ExtR (M, −) → Tor1 (Tn+1 M, −) → 0.

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For an integer n ≥ 0, we say M satisfies (Sn ) if depthRp (Mp ) ≥ min{n, dim(Rp )} for all p ∈ Spec(R). If R is Gorenstein, then M satisfies (Sn ) if and only if ExtiR (Tr M, R) = 0 for all 1 ≤ i ≤ n (see [4, Theorem 4.25]). In particular, M satisfies (S2 ) if and only if it is reflexive, i.e., the natural map M → M ∗∗ is bijective, where M ∗ = HomR (M, R) (see [11, Theorem 3.6]). The following results will be used throughout the paper. Theorem 2.5. Let R be a local complete intersection ring and let M and N be i R–modules. Then TorR i (M, N ) = 0 for all i ≫ 0 if and only if ExtR (M, N ) = 0 for all i ≫ 0. Moreover, if R is a hypersurface, then either pdR (M ) < ∞ or pdR (N ) < ∞. Proof. See [5, Theorem 6.1] and [5, Proposition 5.12].



Theorem 2.6. Let R be a local ring and let M and N be nonzero R–modules. If ExtiR (M, N ) = 0 for all i ≫ 0 and G-dimR (M ) < ∞, then the following statements hold true. (i) If CI-dimR (M ) < ∞, then CI-dimR (M ) = sup{i | ExtiR (M, N ) 6= 0}. (ii) If CI-dimR (N ) < ∞, then G-dimR (M ) = sup{i | ExtiR (M, N ) 6= 0}. Proof. See [1, Theorem 4.2] and [21, Theorem 4.4].



Theorem 2.7. Let R be a local ring, and M , N two R–modules. If CI-dimR (M ) = 0, then ExtiR (M, N ) = 0 for all i > 0 if and only if TorR i (Tr M, N ) = 0 for all i > 0. Proof. First note that CI-dimR (Tr M ) = 0 by [21, Lemma 3.3] and M ≈ Tr Tr M . Now the assertion is clear by [21, Proposition 3.4].  3. Vanishing of Ext for rigid modules We start this section by estimate the Gorenstein dimension of the transpose of M in terms of vanishing of the cohomology modules, ExtiR (M, N ). Lemma 3.1. Let R be a Gorenstein ring and let M and N be nonzero R–modules. Assume that n ≥ 0 is an integer and that the following conditions hold. (1) ExtiR (M, N ) = 0 for all 1 ≤ i ≤ max{1, depthR (N ) − n}. (2) N is rigid. (3) N has reducible complexity. Then G-dimR (Tr M ) ≤ n and TorR i (Tr M, N ) = 0 for all i > 0. Proof. If Tr M = 0, then G-dimR (Tr M ) = 0 and we have nothing to prove so let Tr M 6= 0. As Ext1R (M, N ) = 0, TorR 1 (T2 M, N ) = 0 by the exact sequence (2.4.2). Since N is rigid, we have TorR i (T2 M, N ) = 0 for all i > 0. It follows from the exact sequence (2.4.2) again that Ext1R (M, R) ⊗R N = 0 and since N is nonzero, Ext1R (M, R) = 0. Now it is easy to see that T1 M ≈ ΩT2 M and so TorR i (Tr M, N ) = 0 for all i > 0. Therefore we have the following equality. (3.1.1)

depthR (Tr M ⊗R N ) + depth R = depthR (Tr M ) + depthR (N ),

by Theorem 2.3. Set t = depthR (N ) − n. We argue by induction on t. If t ≤ 1, then depthR (N ) ≤ n + 1. If depthR (N ) = 0, then it is clear that depthR (Tr M ) = depth R by (3.1.1) and so G-dimR (Tr M ) = 0 by Auslander-Bridger formula. Now

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let 0 < depthR (N ) ≤ n + 1. As M ≈ Tr Tr M , we obtain the following exact sequence 0 → Ext1R (M, N ) → Tr M ⊗R N → HomR ((Tr M )∗ , N ) → Ext2R (M, N ), from the exact sequence (2.4.1). As Ext1R (M, N ) = 0, we get the following exact sequence. (3.1.2)

0 → Tr M ⊗R N → HomR ((Tr M )∗ , N ) → Ext2R (M, N ).

Therefore AssR (Tr M ⊗R N ) ⊆ AssR (HomR ((Tr M )∗ , N )) ⊆ AssR (N ), by the exact sequence (3.1.2). Hence depthR (Tr M ⊗R N ) > 0. Now by (3.1.1), it is easy to see that depthR (Tr M ) ≥ depth R − n and so G-dimR (Tr M ) ≤ n. Now suppose that t > 1 and consider the following exact sequence (3.1.3)

0 → ΩM → F → M → 0,

where F is a free R–module. From the exact sequence (3.1.3), we obtain the following exact sequence 0 → M ∗ → F ∗ → (ΩM )∗ → D(M ) → D(F ) → D(ΩM ) → 0. Where D(X) ≈ Tr X for all R–modules X by [4, Lemma 3.9]. As Ext1R (M, R) = 0, we get the following exact sequence (3.1.4)

0 → D(M ) → D(F ) → D(ΩM ) → 0.

Note that D(F ) is free. As ExtiR (ΩM, N ) ∼ = Exti+1 R (M, N ) = 0 for all 1 ≤ i ≤ depthR (N ) − n − 1, we have G-dimR (Tr ΩM ) ≤ n + 1 by induction hypothesis. Therefore, G-dimR (Tr M ) ≤ n by the exact sequence (3.1.4).  Theorem 3.2. Let R be a Gorenstein ring and let M and N be nonzero R–modules such that N has reducible complexity. Assume that N is rigid and that n ≥ 0 is an integer. Then the following statements hold true. (i) If ExtiR (M, N ) = 0 for all 1 ≤ i ≤ max{1, depthR (N ) − n} and M satisfies (Sn ), then G-dimR (M ) = 0. (ii) If ExtiR (M, N ) = 0 for all 1 ≤ i ≤ max{1, depthR (N )−n}, then G-dimR (M ∗ ) ≤ n − 2. Proof. (i). First note that G-dimR (Tr M ) = sup{i | ExtiR (Tr M, R) 6= 0} by [4, Theorem 4.13]. As M satisfies (Sn ), ExtiR (Tr M, R) = 0 for all 1 ≤ i ≤ n by [4, Theorem 4.25]. On the other hand, G-dimR (Tr M ) ≤ n by Lemma 3.1. Therefore G-dimR (Tr M ) = 0 and so G-dimR (M ) = 0 by [4, Lemmm 4.9]. (ii). Note that M ∗ ≈ Ω2 Tr M . By Lemma 3.1, G-dimR (Tr M ) ≤ n and so G-dimR (M ∗ ) ≤ n − 2.  The following is a generalization of [16, Theorem 1]. Corollary 3.3. Let R be a complete intersection and let M and N be nonzero R– modules. Assume that N is a rigid module of maximal complexity. If ExtiR (M, N ) = 0 for all i, 1 ≤ i ≤ max{1, depthR (N ) − 2}, then M ∗ is free. ∗ Proof. By Lemma 3.1, TorR ≈ Ω2 Tr M , i (Tr M, N ) = 0 for all i > 0. As M R ∗ ∗ Tori (M , N ) = 0 for all i > 0 and so cxR (M )+cxR (N ) ≤ codim R by [5, Theorem II]. Since N has maximal complexity, it follows that cxR (M ∗ ) = 0. Therefore, pdR (M ∗ ) = G-dimR (M ∗ ) = 0 by Theorem 3.2(ii). 

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It is well-known that over a regular local ring every finite module is rigid. In the following we collect some other examples of rigid modules. Example 3.4. (i) A class of rigid modules was discovered by Peskine and Szpiro [19]. They proved that if R is local, and the minimal free resolution of M over R is of the form 0 → Rm → Rk+m → Rk → 0, for some m > 0 and k > 0, then M is rigid. In [22], Tchernev discovered a new class of rigid modules. He showed that if R is local, and the minimal free resolution of M over R is of the form 0 → Rk → Rm+1 → Rm → 0, for some m > 0 and k > 0, then M is rigid ([22, Theorem 3.6]). (ii) Let R be an admissible hypersurface with isolated singularity and let N be an R–module. If [N ] = 0 in G(R)Q , then N is rigid [10, Corollary 4.2]. b = S/(f ) where (S, n) (iii) Let (R, m) be a local hypersurface ring such that R is a complete unramified regular local ring and f is a regular element of S contained in n2 . Let M be an R-module of finite projective dimension. Then M is rigid [17, Theorem 3]. In the following, we generalize [15, Corollary 1]. Theorem 3.5. Let R be a local complete intersection ring and let M and N be nonzero R–modules. Assume the following conditions hold. (i) N is rigid. (ii) M satisfies (Sn ) for some n ≥ 0. (iii) ExtiR (M, N ) = 0 for some positive integer i such that i ≥ depthR (N ) − n. Then CI-dimR (M ) = sup{j | ExtjR (M, N ) 6= 0} < i. Proof. Set L = Ωi−1 M . Note that L satisfies (Sn+i−1 ) and Ext1R (L, N ) = 0. Now by Theorem 3.2(i), CI-dimR (L) = G-dimR (L) = 0. By Lemma 3.1, TorR j (Tr L, N ) = j 0 for all j > 0 and so ExtR (L, N ) = 0 for all j > 0 by Theorem 2.7. Therefore ExtjR (M, N ) = 0 for all j ≥ i and so CI-dimR (M ) = sup{j | ExtjR (M, N ) 6= 0} < i by Theorem 2.6.  The following is a generalization of [15, Corollary 2] Theorem 3.6. Let R be a local complete intersection ring and let M and N be nonzero R–modules. Suppose that N is rigid and that M satisfies (Sn ) for some n ≥ 0. If depthR (N )−n ≤ CI-dimR (M ), then for all i > 0 in the range depthR (N )−n ≤ i ≤ CI-dimR (M ), we have ExtiR (M, N ) 6= 0. Proof. If ExtiR (M, N ) = 0 for some depthR (N ) − n ≤ i ≤ CI-dimR (M ), then Ext1R (Ωi−1 M, N ) ∼ = ExtiR (M, N ) = 0. Note that Ωi−1 M satisfies (Sn+i−1 ). Now by Theorem 3.2(i), we have CI-dimR (Ωi−1 M ) = G-dimR (Ωi−1 M ) = 0. Therefore CI-dimR (M ) < i by [6, Lemma 1.9], which is a contradiction.  Let R be a hypersurface and let M and N be R–modules such that lengthR (N ) < ∞. It is well-known that if ExtiR (M, N ) = 0 for some i > CI-dimR (M ), then ExtnR (M, N ) = 0 for all n > CI-dimR (M ) (see for example [8, Corollary 3.5]). In special cases, we can remove the condition that i > CI-dimR (M ).

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b = S/(f ) where Corollary 3.7. Let (R, m) be a local hypersurface ring such that R (S, n) is a complete unramified regular local ring and f is a regular element of S contained in n2 . Let M and N be nonzero R-modules such that lengthR (N ) < ∞. If ExtnR (M, N ) = 0 for some n ≥ 1, then the following statements hold true. (i) CI-dimR (M ) = sup{i | ExtiR (M, N ) 6= 0} < n. (ii) either pdR (M ) < ∞ or pdR (N ) < ∞. Proof. First note that N is rigid by [13, Theorem 2.4]. It follows from Theorem 3.5 that CI-dimR (M ) = sup{i | ExtiR (M, N ) 6= 0} < n. As ExtiR (M, N ) = 0 for all i ≫ 0, either pdR (M ) < ∞ or pdR (N ) < ∞ by Theorem 2.5.  As an application of Theorem 3.2, we have the following result. Corollary 3.8. Let R be an admissible hypersurface and let M and N be nonzero R–modules such that cxR (N ) = 1. Assume that the minimal free resolution of N is eventually periodic of period one and that M satisfies (Sn ) for some n ≥ 0. Then the following statements hold true. (i) If depthR (N )−n ≤ CI-dimR (M ), then for all i > 0 in the range depthR (N )− n ≤ i ≤ CI-dimR (M ), we have ExtiR (M, N ) 6= 0. (ii) If ExtiR (M, N ) = 0 for some positive integer i such that i ≥ depthR (N ) − n, then pdR (M ) < i. (iii) If ExtiR (M, N ) = 0 for all i, 1 ≤ i ≤ max{1, depthR (N ) − 2}, then M ∗ is free. Proof. Note that N is rigid by [10, Corollary 5.6]. Now the first assertion is clear by Theorem 3.6. (ii). By Theorem 3.5, ExtjR (M, N ) = 0 for all j ≥ i. Therefore, pdR (M ) < ∞ by Theorem 2.5 and so pdR (M ) < i. (iii). Note that N has maximal complexity. Therefore, the assertion is clear by Corollary 3.3.  Let R be an admissible hypersurface with isolated singularity of dimension d > 1. By [10, Theorem 3.4], every R–module of dimension less than or equal one is rigid. As an immediate consequence of Theorem 3.5, we have the following result. Corollary 3.9. Let R be an admissible hypersurface with isolated singularity of dimension d > 1 and let M and N be nonzero R–modules such that dimR (N ) ≤ 1. If ExtnR (M, N ) = 0 for some n > 0, then CI-dimR (M ) = sup{i | ExtiR (M, N ) 6= 0} < n. Moreover, either pdR (M ) < ∞ or pdR (N ) < ∞. In the dimension 2 case, we have the following result. Proposition 3.10. Let R be an admissible hypersurface of dimension 2. Assume further that R is normal. Let M and N be nonzero R–modules such that depthR (N ) ≤ depthR (M ) + 1. If Ext1R (M, N ) = 0, then CI-dimR (M ) = 0 and ExtiR (M, N ) = 0 for all i > 0. Moreover, either M is free or N has finite projective dimension. Proof. First note that N is rigid by [10, Corollary 3.6]. If depthR (N ) ≤ 1, then the assertion is clear by Theorem 3.5. Now let N be maximal Cohen-Macaulay. Then depthR (M ) > 0 and so (3.10.1)

CI-dimR (M ) = sup{i | ExtiR (M, R) 6= 0} = 2 − depthR (M ) ≤ 1.

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R By Theorem 2.4, TorR 1 (T2 M, N ) = 0. As N is rigid, Tori (T2 M, N ) = 0 for all 1 i > 0. It follows from Theorem 2.4 again that ExtR (M, R) = 0 and so M is maximal Cohen-Macaulay by (3.10.1). Now it is easy to see that Tr M ≈ ΩT2 M i and so TorR i (Tr M, N ) = 0 for all i > 0. Therefore, ExtR (M, N ) = 0 for all i > 0 by Theorem 2.7 and so either M is free or N has finite projective dimension by Theorem 2.5. 

4. Vanishing of Ext over complete intersection rings Let R be a local complete intersection ring of codimension c and let M and N be R R–modules. In [18], Murthy proved that if TorR n (M, N ) = Torn+1 (M, N ) = · · · = R TorR n+c (M, N ) = 0 for some n > 0, then Tori (M, N ) = 0 for all i ≥ n. It is easy to see that a similar statement is not true in general, with Tor replaced by Ext. In the following, we prove a similar result for Ext with an extra hypothesis. The following result is a generalization of [14, Corollary]. Theorem 4.1. Let R be a local complete intersection ring of codimension c and let M and N be nonzero R–modules. Assume n is a positive integer. If ExtiR (M, N ) = 0, for all i, n ≤ i ≤ n + c and depthR (N ) ≤ n + c, then CI-dimR (M ) = sup{i | ExtiR (M, N ) 6= 0} < n. Proof. Without lose of generality we may assume that R is complete. We have R = Q/(x) with Q a complete regular local ring and x an Q-sequence of length c contained in the square of the maximal ideal of Q. We argue by induction on c. If c = 0, then R is a regular local ring and so pdR (M ) = sup{i | ExtiR (M, N ) 6= 0} < n by [15, Corollary 1]. For c > 0, set S = Q/(x1 , . . . , xc−1 ). Therefore R ∼ = S/(xc ). Note that depthR (N ) = depthS (N ). The change of rings spectral sequence (see [20, Theorem 11.66]) ExtpR (M, ExtqS (R, N )) ⇒ Extp+q S (M, N ) p

degenerates into a long exact sequence i+1 · · · → ExtiR (M, N ) → ExtiS (M, N ) → Exti−1 R (M, N ) → ExtR (M, N ) → · · · .

It follows that ExtiS (M, N ) = 0 for all i, n + 1 ≤ i ≤ n + c, and so by induction hypothesis we conclude that CI-dimS (M ) = sup{i | ExtiS (M, N ) 6= 0} < n + 1. i+1 ∼ Therefore, Exti−1 R (M, N ) = ExtR (M, N ) for all i > n. As c > 0, it is clear that ExtiR (M, N ) = 0 for all i ≥ n and so CI-dimR (M ) = sup{i | ExtiR (M, N ) 6= 0} < n by Theorem 2.6.  In special cases, one can improve the Theorem 4.1 slightly. The following is a generalization of corollary 3.7. b = S/(f ) where (S, n) is Proposition 4.2. Let (R, m) be a local ring such that R a complete unramified regular local ring and f = f1 , f2 , . . . , fc is a regular sequence of S contained in n2 . Assume that n ≥ 0 is an integer and that M and N are nonzero finite R-modules such that lengthR (N ) < ∞. If ExtiR (M, N ) = 0 for all i, n + 1 ≤ i ≤ n + c, then CI-dimR (M ) = sup{i | ExtiR (M, N ) 6= 0} ≤ n. Proof. Without lose of generality we may assume that R is complete and R = S/(f ) where (S, n) is a complete unramified regular local ring and f = f1 , f2 , . . . , fc is a regular sequence of S contained in n2 . We argue by induction on c. If c = 1,

10

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then the assertion holds by Corollary 3.7. For c > 1, set Q = S/(f1 , . . . , fc−1 ). Therefore, R ∼ = Q/(fc ). Note that lengthQ (N ) < ∞. The change of rings spectral sequence ExtpR (M, ExtqQ (R, N )) ⇒ Extp+q Q (M, N ) p

degenerates into a long exact sequence i+1 · · · → ExtiR (M, N ) → ExtiQ (M, N ) → Exti−1 R (M, N ) → ExtR (M, N ) → · · · .

It follows that ExtiQ (M, N ) = 0 for all i, n + 2 ≤ i ≤ n + c, and so by induction hypothesis we conclude that CI-dimQ (M ) ≤ n + 1 and ExtiQ (M, N ) = 0 for all i+1 ∼ i > n + 1. Therefore, Exti−1 R (M, N ) = ExtR (M, N ) for all i > n + 1. As c > 1, i it is clear that ExtR (M, N ) = 0 for all i > n and so CI-dimR (M ) = sup{i | ExtiR (M, N ) 6= 0} ≤ n by Theorem 2.6.  As an application of Theorem 4.1, we can generalize [15, Corollary 1] as follows. Corollary 4.3. Let R be a local complete intersection ring of codimension c and let M and N be nonzero R–modules. Assume that n > 0 and t ≥ 0 are integers and that the following conditions hold. (i) ExtiR (M, N ) = 0 for all i, n ≤ i ≤ n + c. (ii) M satisfies (St ). (iii) depthR (N ) ≤ n + c + t. Then CI-dimR (M ) = sup{i | ExtiR (M, N ) 6= 0} < n. Proof. We argue by induction on t. If t = 0, then the assertion is clear by Theorem 4.1. Now suppose that t > 0 and consider the universal pushforward of M , (4.3.1)

0 → M → F → M1 → 0,

where F is free. It is easy to see that M1 satisfies (St−1 ). From the exact sequence (4.3.1), it is clear that (4.3.2) Exti (M, N ) ∼ = Exti+1 (M1 , N ) for all i > 0. R

R

Therefore, ExtiR (M1 , N ) = 0 for all i, n+1 ≤ i ≤ n+c+1. By induction hypothesis, we conclude that ExtiR (M1 , N ) = 0 for all i > n. By (4.3.2), ExtiR (M, N ) = 0 for all i ≥ n and so CI-dimR (M ) = sup{i | ExtiR (M, N ) 6= 0} < n by Theorem 2.6.  Acknowledgements. I would like to thank Olgur Celikbas and my thesis adviser Mohammad Taghi Dibaei for valuable suggestions and comments on this paper. References 1. T. Araya and Y. Yoshino, Remarks on a depth formula, a grade inequality and a conjecture of Auslander, Comm. Algebra 26 (1998), 3793-3806. 5 2. M. Auslander, Anneaux de Gorenstein, et torsion en alg` ebre commutative, in: S´ eminaire d’Alg` ebre Commutative dirig´ e par Pierre Samuel, vol. 1966/67, Secr´ etariat math´ ematique, Paris, 1967. 3 3. M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631-647. 1, 4 4. M. Auslander and M. Bridger, Stable module theory, Mem. of the AMS 94, Amer. Math. Soc., Providence 1969. 3, 4, 5, 6 5. L. L. Avramov, R.-O. Buchweitz, Support varieties and cohomology over complete intersections. Invent. Math. 142 (2000): 285-318. 5, 6

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L. L. Avramov, V. N. Gasharov and I. V. Peeva, Complete intersection dimension, Publ. Math. I.H.E.S. 86 (1997), 67-114. 3, 4, 7 7. P. Bergh, Modules with reducible complexity, J. Algebra 310 (2007), 132-147. 3 8. P. Bergh, On the vanishing of homology with modules of finite length, Math. Scand. to appear. 7 9. P. Bergh and D. Jorgensen, The depth formula for modules with reducible complexity. Illinois J. Math. to appear. 3, 4 10. H. Dao, Decent intersection and Tor-rigidity for modules over local hypersurfaces, Transactions of the AMS, to appear. 7, 8 11. E. G. Evans and P. Griffith. Syzygies, volume 106 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1985. 5 12. T. H. Gulliksen, A change of rings theorem, with applications to Poincare series and intersection multiplicity, Math. Scand. 34 (1974), 167-183. 2 13. C. Huneke and R. Wiegand, Tensor products of modules and the rigidity of Tor, Math. Ann. 299 (1994), 449-476. 4, 8 14. P. Jothilingam, Test modules for projectivity. Proc. Amer. Math. Soc. 94 (1985), 593-596. 9 15. P. Jothilingam, Syzygies and Ext. Math. Z. 188 (1985), 278-282. 1, 2, 7, 9, 10 16. P. Jothilingam and T. Duraivel, Test Modules for Projectivity of Duals, Comm. Alg. 38:8 (2010), 2762-2767. 1, 2, 6 17. S. Lichtenbaum, On the vanishing of Tor in regular local rings, Ill. J. Math. 10 (1966), 220-226. 1, 7 18. M. P. Murthy, Modules over regular local rings, Illinois J. Math. 7 (1963), 558-565. 9 19. C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, Inst. Hautes ` Etudes Sci. Publ. Math. 42 (1973), 47119. 7 20. J. Rotman, An Introduction to Homological Algebra. Academic Press, New York (1979). 9 21. A. Sadeghi, A note on the depth formula and vanishing of cohomology. preprint, 2012. 5 22. A. Tchernev, Free direct summands of maximal rank and rigidity in projective dimension two. Comm. Alg. 34:2 (2006), 671-679. 7 School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran. E-mail address: [email protected]